91影视

We have given that system of equation is $$\begin{gathered} {\text{y}}_1{\text{= 2x}}_2 \\ {\text{y}}_2{\text{= 3x}}_3 \\ {\text{y}}_3{\text{=x}}_1 \end{gathered}$$.

For the transformation of ${\mathrm{鈩�}}^3$to ${\mathrm{鈩�}}^3$it can be written as

$\begin{array}{rcl}T(x_1,x_2,x_3)& =& (y_1,y_2,y_3)\\ T(x_1,x_2,x_3)& =& (2x_2,3x_3,x_1)\\ & & \end{array}$

A transformation from$R^m鈫�R^n$ is said to be linear if the following condition holds:

1. Identity of${\mathrm{鈩�}}^m$should be mapped to identity of${\mathrm{鈩�}}^n$ .
2. $T(a+b)=T\left(a\right)+T\left(b\right)$For all$a,b鈭�R^m$ .
3. $T\left(ca\right)=cT\left(a\right)$Where c is any scalar and$a鈭�R^m$ .

Identity of $R^3$is $(0,0,0)$ .

Now we will find the transformation of identity element.

$T(0,0,0)=\left(2\right(0),0,0)=(0,0,0)$

Now let $a=(u_1,u_2,u_3),b=(v_1,v_2,v_3)鈭�R^3$.

Then the transformation,

$\begin{array}{rcl}T(a+b)& =& T\left(\right(u_1,u_2,u_3)+(v_1,v_2,v_3\left)\right)=T(u_1+v_1,u_2+v_2,u_3+v_3)\\ & 鈬�& T(u_1+v_1,u_2+v_2,u_3+v_3)=\left(2\right(u_2+v_2),3(u_3+v_3),u_1+v_1)\\ & 鈬�& T(u_1+v_1,u_2+v_2,u_3+v_3)=(2u_2+2v_2,3u_3+3v_3,u_1+v_1)\\ & 鈬�& T(u_1+v_1,u_2+v_2,u_3+v_3)=(2u_2,3u_3,u_1)(2v_2,3v_3,v_1)=T\left(a\right)+T\left(b\right)\\ & 鈬�& T(a+b)=T\left(a\right)+T\left(b\right)\end{array}$

Now for condition 3 Let $a=(u_1,u_2,u_3)鈭�R^3$and $c$be any scalar.

$\begin{array}{rcl}T\left(ca\right)& =& T\left(c\right(u_1,u_2,u_3\left)\right)\\ & =& (2c{u}_2,3c{u}_3,c{u}_1)\\ & =& c(2u_2,3u_3,u_1)\\ & =& cT\left(a\right)\\ & 鈬�& T\left(ca\right)=cT\left(a\right)\\ & & \end{array}$

Thus, all the conditions are true, so the given transformation is a linear transformation.